Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parallel solvers for virtual element discretizations of elliptic equations in mixed form

F. Dassi, S. Scacchi

TL;DR

It is shown that the proposed VEM discretization recovers the expected theoretical convergence properties and the performance of the direct and iterative parallel solvers taken into account is analized.

Abstract

The aim of this paper is twofold. On the one hand, we test numerically the performance of mixed virtual elements in three dimensions for the first time in the literature to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed VEM discretization recovers the expected theoretical convergence properties and we analize the performance of the direct and iterative parallel solvers taken into account.

Parallel solvers for virtual element discretizations of elliptic equations in mixed form

TL;DR

It is shown that the proposed VEM discretization recovers the expected theoretical convergence properties and the performance of the direct and iterative parallel solvers taken into account is analized.

Abstract

The aim of this paper is twofold. On the one hand, we test numerically the performance of mixed virtual elements in three dimensions for the first time in the literature to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed VEM discretization recovers the expected theoretical convergence properties and we analize the performance of the direct and iterative parallel solvers taken into account.

Paper Structure

This paper contains 17 sections, 7 theorems, 59 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Polynomial spaces and bases
  3. Polynomial decomposition
  4. Basis for $\mathcal{G}_k(P)$ and $\GPerp$
  5. Basis for $\mathcal{G}_k(P)$
  6. Basis for $\GPerp$
  7. Model problem: elliptic equation in mixed form
  8. Virtual element discretization
  9. Definition of $\boldsymbol{V}_{k,h}(\Omega_h)$
  10. Definition of $Q_{k-1,h}(\Omega_h)$
  11. The discrete local forms
  12. Numerical results
  13. Parallel preconditioners
  14. Numerical results
  15. Test 1: strong scaling
  16. ...and 2 more sections

Key Result

Proposition 2.1

Considering a vectorial scaled monomial with only the first component different from 0, Equation eqn:decomp becomes where

Figures (6)

  • Figure 1: A sample of the mesh taken into account.
  • Figure 2: Convergence lines for each set of meshes taken into account and degrees $k=2,3$ and 4.
  • Figure 3: Strong scaling test, Cube meshes. GMRES iterations (left) and solution time (right) as a function of the number of procs for $k=2$ (first row) and $k=3$ (second row). In the solution time plots, the black dashed line indicates the steepness of the ideal time reduction.
  • Figure 4: Strong scaling test, Octa meshes. Same format as in Fig. \ref{['fig_hexa_scal']}.
  • Figure 5: Strong scaling test, CVT meshes. Same format as in Fig. \ref{['fig_hexa_scal']}.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.4
  • Corollary 2.1
  • Remark 2.3
  • Proposition 2.5
  • Proposition 2.6
  • ...and 3 more